\section{Methodology}
\label{sec:risk.meth}

To validate our findings, we carried out comprehensive simulations
over a wide range of networks, listed in Table \ref{tab:networks}. For
each type of network, we ran simulations on different parameters
$\rb{\pt, \pb, \pf, \pv}$, to confirm that our findings are
ubiquitous. Because simulations are stochastic processes, we run ten
iterations for each set of parameters $\rb{\pt, \pb, \pf, \pv}$ on a
given graph, and then average over the epidemic severities. Figure
\ref{fig:scalefree} shows typical graphs we get across all networks we
have done simulations on. In Figure \ref{fig:scalefree} we omitted the
95 percent confident interval because the epidemic severity is tightly
concentrated around the mean.  We refer the reader to supplementary
information for the complete simulation results.

\junk{
The simulation begins with a contact graph $G$ of size $n$. The program goes over all the nodes, and vaccinates each with probability $\pv$. With probability $\pv(1-\pf)$, a node will take vaccine and succeed. In this case, the program removes this node from $G$. With probability $\pv\pf$ a node will take vaccine but fail. In this case, the program keep this node in $G$ and mark it as ``vaccination failure''. With probability $1-\pv$, a node will not take vaccine. In this case, the program mark it as ``unvaccinated''. Then, the program goes through all the edges incident on the residual nodes.
}
\begin{table}[ht]
\caption{Descriptions of the networks used in the paper. For each network we show its type, name, number of nodes $n$ and edges $m$.}
\label{tab:networks}
\begin{center}
\begin{tabular}{p{1in} p{1.2in} r r p{2.4in}}
\hline
\multicolumn{2}{c}{name} & $n$ & $m$ & description \\
\hline
Human contact & NewRiverValley \cite{wsc09pop} & 74,375 & 1,888,833 & Synthetic human contact network for New River Valley county in Virginia. \\
Social communication & Enron mail \cite{enron, enron-link} & 36,691 & 367,666 & Email communication network in a company. \\
Peer-to-peer network & Gnutella \cite{ripeanu+fi:p2p, p2p-link} & 10,876 & 39,994 &Gnutella peer-to-peer file sharing network from August 2002 \\
Random graphs & Preferential attachment \cite{barabasi+perferattach99} & 100,000 & 300,000  & Generated using Python NetworkX library. \\
% & Erd\"{o}s and R\'{e}nyi \cite{erdos1960} & 100,000 & 5,000,000 & \\
\hline
\end{tabular}
\footnotetext[1]{foot note}
\end{center}
\end{table}
